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Tabu-search simulation optimizationapproach for flow-shop schedulingwith multiple processors — a casestudyT. Yang , Y. Kuo & I. Changa Institute of Manufacturing Engineering , National ChengKung University , Tainan 701, Taiwan, Republic of Chinab Institute of Manufacturing Engineering , National ChengKung University , Tainan 701, Taiwan, Republic of China E-mail:Published online: 21 Feb 2007.

To cite this article: T. Yang , Y. Kuo & I. Chang (2004) Tabu-search simulation optimizationapproach for flow-shop scheduling with multiple processors — a case study, InternationalJournal of Production Research, 42:19, 4015-4030, DOI: 10.1080/00207540410001699381

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int. j. prod. res., 1 october 2004,vol. 42, no. 19, 4015–4030

Tabu-search simulation optimization approach for flow-shop scheduling

with multiple processors — a case study

T. YANG*, Y. KUO and I. CHANG

The flow shop with multiple processors (FSMP) environment is relativelycommon and has a variety of applications. The majority of academic authorssolve the scheduling problem of FSMP using deterministic data that ignore thestochastic nature of a real-world problem. Discrete-event simulation can model anon-linear and stochastic problem and allows examination of the likely behaviourof a proposed manufacturing system under selected conditions. However, it doesnot provide a method for optimization. The present paper proposes to solve theFSMP scheduling problem by using a tabu-search simulation optimizationapproach. It features both the stochastically modelling capability of thediscrete-event simulation and the efficient local-search algorithm of tabu search.A case study from a multilayer ceramic capacitor manufacturing illustrates theproposed solution methodology. Empirical results show promise for the practicalapplication of the proposed methodologies. Future research opportunities arethen addressed.

1. Introduction

The flow shop with multiple processors (FSMPs) is also known as ‘flexible flowline’ and ‘hybrid flow shop’. The FSMP scheduling problem involves the sequencingof jobs in a flow shop in which, at any stage, more than one identical machinemight exist. A machine can process exactly one job at a time, and the jobs are subjectto precedence constraints (Salvador 1973). This type of scheduling environmentis relatively common and has a variety of applications — including semiconductorand electronics manufacturing, and petrochemical production (Santos et al. 1996,Botta-Genoulaz 2000). First-in-first-out (FIFO) is used as the sequencing procedurein the subsequent stages because it is the strategy usually used in FSMP environments(Pinedo 1995).

The FSMP scheduling problem is NP-hard and its optimal solution is computa-tionally prohibitive for a practical-size problem (Gupta 1988, Brah 1992). In the pastfew decades there has been a significant number of reports in the literature that havediscussed the single-processor flow-shop scheduling problem (Campbell et al. 1970,Townsend 1977, Nawas et al. 1983, Proust et al. 1991). Studies on FSMP schedulingproblems are relatively recent. Of these, most have dealt with the makespan criterionand have usually been limited to two stages (Gupta 1988, Gupta and Tunc 1991,Deal et al. 1994, Uetake et al. 1995). The two-stage problem is a special case of theFSMP scheduling problem and is not practical.

Revision received March 2004.Institute of Manufacturing Engineering, National Cheng Kung University, Tainan 701,

Taiwan, Republic of China.*To whom correspondence should be addressed. e-mail: tyang@mail.ncku.edu.tw

International Journal of Production Research ISSN 0020–7543 print/ISSN 1366–588X online # 2004 Taylor & Francis Ltd

http://www.tandf.co.uk/journals

DOI: 10.1080/00207540410001699381

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The majority of reports in the literature solve the FSMP scheduling problemusing deterministic data that ignore the stochastic nature of a real-world problem— such as process, set-up, load/unload, machine failure times, etc. Discrete-eventsimulation can model a non-linear and stochastic problem and allows examinationof the likely behaviour of a proposed manufacturing system under selected condi-tions. However, it does not provide a method for optimization (Yang and Tseng2002).

Optimization-based simulation in solving an FSMP scheduling problem is rarein the literature. Grangeon et al. (1999) used a generic FSMP simulation model toevaluate the performances of an FSMP system using different dispatching rules.Hsieh et al. (2001) solved the semiconductor wafer fabrication-scheduling problemusing rule-based simulation. Jeong (2000) proposed a conceptual framework forthe development of optimized simulation-based scheduling systems. This frameworkintegrated a discrete-event simulator and a rule-based system to handle condition-based events for a given environment. Azzaro-Pantel et al. (1998) solved a batch–plant-scheduling problem using an iterative discrete-event simulation and geneticalgorithm methodology.

Tabu search (TS) is a local search-based optimization method that has beensuccessfully applied to solve many difficult combinatorial optimization problems.Its applications to sequencing-type problems are impressive and have exhibitedconsiderable robustness (Ben-Daya and Al-Fawzan 1998). For many schedulingoptimization problems, the empirical results from Taillard (1989), Widmerand Hertz (1989), Barnes and Laguna (1993) and Dorn et al. (1996) showed thatTS can outperform simulated annealing, branch-and-bound method, genetic algo-rithms and random search methods. Although the comparison is only possiblebetween implementations of metaheuristics for a given problem, the potential ofTS for the present study is clear by examining the empirical results from existingliterature.

The present paper proposes to solve the FSMP scheduling problem by a TSsimulation optimization approach. It explores the hybrid TS and discrete-eventsimulation methodologies in solving a complex manufacturing system problem,which is not well addressed in literature. It features both the stochastic modellingcapability of the discrete-event simulation and the efficiency of the TS algorithm.In addition, a practical case study from a multilayer ceramic capacitor (MLCC)manufacturing is used to illustrate the proposed solution methodology. For thistype of problem, the case study presents the MLCC manufacturing application forits first time in literature.

2. Case study

Passive components are used in all electronics to regulate the flow of electricityand to store electric charge for peak power needs, for frequency control and forfiltering. Among the passive components, capacitors (mostly tantalum capacitorsand MLCCs) account for approximately 70% of the USA’s revenue. The use oftantalum capacitors and MLCCs is growing quickly because they are surface moun-table and are used primarily in communication and computing (Harris and Roesch2001).

The MLCC manufacturing process begins from ceramic powder preparationand ends with reel taping. Figure 1 illustrates a generic MLCC manufacturingprocess flow. Figure 2 illustrates the appearance of an MLCC. The dimensions

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for a fine-pitch MLCC are about 1.0, 0.5 and 0.5mm for length, width and height,respectively.

The MLCC manufacturing has a sole process flow with precedence constraints.In addition, there are multiple machines in each stage. Thus, it is an FSMP problem.However, this is a more complex modelling problem than is a conventional FSMPproblem for the following reasons. First, it has more than 10 stages in its processflow. Second, the binder burnout and sintering stages are batch-processing proces-sors. Finally, its product mix is determined by both material type and product size,the combination of which decides the set-up requirements.

The case study presented here is of a leading MLCC manufacturing companylocated in Tainan, Taiwan. Its annual sales are expected to exceed US$229 millionin 2003. An MLCC plant usually has different inventory-carry policies in the before-stacking and after-testing stages. The raw materials in the before-stacking stage areprimarily powder and foil, and are received in bulk. Therefore, batch production is

Figure 1. Generic MLCC process flow.

Figure 2. A MLCC example.

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a common practice and this results in highly variable work-in-process inventory. The

taping stage usually uses a late customization policy and is not part of the process

flow. Thus, this research models the case study problem from the stacking stage to

the testing stage. In total, the case study problem entails nine stages. The detailed

modelling data — including stage names, number of machines in each stage, set-up

time, machine failure information and processing time — are shown in appendices

A1–5. The present study has modified some data to protect proprietary information

from the case study company.

There are 24 product types consisting of the various combinations of six material

types and four product sizes. The smallest processing unit is termed a ‘lot’. The lot

size is product specific and the various sizes are collected in appendix A2. Due to

the batch-processing requirements of the binder burnout and sintering stages, the

study assigns a batch size of six lots for experimental purposes. Hereafter, each

customer order represents one batch. This study adopts the FIFO method for the

interstage job sequencing. It assumes that all production lots are available before

the production starts, and then tries to decide the job sequence for entering the

production line. The proposed methodology solves the order-sequencing problem

for MLCC manufacturing and thus minimizes the total tardiness that is one of the

key performance indices of the case study.

3. Proposed methodology

The proposed TS simulation optimization approach is an iterative search algo-

rithm. It follows the efficient TS procedure and uses computer simulation to evaluate

system performance for each search iteration.

The key requirement of any global optimization method is that it should be able

to avoid entrapment in seeking a locally optimal solution while continuing the search

to provide a near-optimal final solution. TS has been proposed by Glover (1989) as

an iterative process that explores the solution space by moving from one solution

to another neighbouring solution. By using the short-term memory of recent solu-

tions, the optimization routines can effectively escape from local optimal solutions

(Glover and Laguna 1993).

TS begins with an initial solution, and then moves from one solution S to another

solution S’, which is located in its neighbourhood N(S ). A solution S’, which is

worse than the current solution S, might be accepted to escape from the entrapment

of a local optimum. The most recent moves are classified as tabu for a particular num-

ber of iterations — i.e. tabu tenure or tabu list size — to avoid the cyclic searching

path. Aspiration criterion is used when a move is in tabu list, but it has a better

solution. In this instance, the move is released from the tabu list. The search will

terminate when it satisfies the stopping criterion. When TS uses the above short-term

flexible memory structures, it is usually referred to as a ‘simple TS’ (Glover 1989),

which is hereafter denoted as ‘TS1’.

A more advanced type of TS includes frequency-based memory intensification

and frequency-based memory diversification. The former seeks to reinforce moves

that incorporate attributes of good solutions found in the past, whereas the latter

seeks to drive the search into unexplored regions (Wen and Huang 1996). The TS

with intensification and diversification strategies will hereafter be denoted by ‘TS2’.

Details of the proposed methodology are discussed below.

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3.1. Initial solutionAn initial solution to solving a scheduling-type problem often involves, a priori,

the use of a heuristic approach, and this often influences the quality of the finalsolution. In the case study under consideration here, the earliest-due-date (EDD)rule is used to generate the initial solution. EDD is an effective rule for minimizingproduction tardiness in most instances (Hopp and Spearman 2001).

3.2. MoveFor TS operation, a neighbourhood solution can be reached from the current

solution. The pairwise-exchange (or swap) method is often used as a move to con-struct a neighbourhood solution in a permutation-type problem (Glover et al. 1995).This study randomly exchanges two orders to reach a neighbourhood solution. Theunderlying rationale for a random exchange is to avoid entrapment in local optimumand to have the chance to explore a different solution search area.

3.3. Neighbourhood sizeThe neighbourhood size is the number of candidate solutions to be evaluated in

each iteration along the search process. The existing literature discussed three dif-ferent neighbour selection strategies (Ben-Daya and Al-Fawzan 1998):

. Choose the first neighbourhood solution that improves the current solution(referred to as ‘first-fit strategy’ hereafter).

. Consider a subset of neighbours and search the best solution.

. Search the whole neighbourhood and choose the best solution.

In general, it is computationally too expensive to explore the whole neighbourhoodof the current solution. A subset of neighbours is often used to alleviate the compu-tational efforts (Alvarez-Valdes et al. 2002). In addition, the size of the subset is toolarge to be useful (Higgins 2001, Negenman 2001). In other words, the improvementto solution quality will become negligible when the size of the subset exceeds acertain level. The larger the subset size, the longer the computational time.When the computational efficiency is a concern, the first-fit strategy can be used(Ben-Daya and Al-Fawzan 1998). In this research, we adopted the first-fit strategyin consideration of the computational burden associated with a simulationapproach.

3.4. Performance evaluationA discrete-event simulator capable of modelling a complicated stochastic prob-

lem is used to evaluate the performance of each TS move. A commercial softwareprogram, eM-Plant (Tecnomatix Technologies 2000), was used as the simulationtool. The TS algorithms were coded using the eM-plant embedded programminglanguage (called Method) for customized applications. Its syntax is similar to theprogramming language Cþþ.

Due to the embedded random variations in a discrete-event simulator, a certainnumber of replications might be required to obtain an adequate confidence intervalfor estimating the population mean. A smaller confidence interval is a result of anincreased number of replications. However, there is a trade-off between an improvedconfidence interval and computing time. Thus, we perform a preliminary study toinvestigate the adequate number of replications, as described below.

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Five production scenarios are generated (table 1). Those scenarios represent awide range of product mixes. In addition, the due date for each batch is randomlygenerated using a uniform distribution, as shown in the far-right column of table 1.Each scenario uses the same sequence from the EDD rule, but collates the outputs(tardiness values) from different numbers of replications. The result of each replica-tion represents one sample. Then, the coefficient of variation (CV), which is defined asthe ratio of the sample standard deviation to the sample mean, is used as an indicatorof the magnitude of variance. Figure 3 shows the relationship between CV andthe number of replications for each of those five production scenarios definedin table 1.

Figure 3 shows minor CVs (<0.1) in most instances. A system is considered tobe stable when its CV<0.75 (Hopp and Spearman 2001). In view of these results andtaking into account computational efficiency, four replications are taken as beingadequate for the case study.

3.5. Aspiration criterionThe proposed TS algorithm searches only for non-tabu moves except it exhausts

all the non-tabu moves and cannot find an improved solution that is better thanthe current solution. If the best search result from tabu list outperforms the bestsolution found so far, then the aspiration criterion overrides the tabu rule. The pro-posed search scheme has been adopted by Yang et al. (1999) to generate promisingresults.

Production scenario:

0.000.020.040.060.08

0.100.120.140.16

1 2 3 4 5 6 7 8 9

Number of replication

CV

1 2 3 4 5

Figure 3. CV analysis.

Scenarionumber

Number ofbatches

Batch size(Lot)

Number ofmaterial types

Number ofproduct sizes

Due datedistribution

(days)

1 9 6 3 3 UNIF (8, 14)2 12 6 4 3 UNIF (8, 19)3 16 6 4 4 UNIF (8, 26)4 20 6 5 4 UNIF (8, 31)5 24 6 6 4 UNIF (8, 35)

Table 1. Production scenarios.

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3.6. Intensification and diversification

TS2 uses a long-term memory structure — intensification and diversification —

to generate a new initial search sequence, and then restarts the TS procedure. Themethodology proposed by Ben-Daya and Al-Fawzan (1998) is adapted for the long-

term memory as follows.

The long-term memory starts with an n�m frequency matrix in which the ncolumns and the m rows represent the n positions in a sequence and the m produc-

tion batches, respectively. Note that n¼m for this study. Each entry in the matrix is

denoted as fij and is incremented by 1 whenever batch i visits position j. Let k be anindex for the iteration number. Then, the proposed intensification and diversification

method has five steps as follows:

Step 0. Let k¼ 1.

Step 1. Find the entry fij of the frequency matrix having the largest value.

Step 2. Assign order i to position j.Step 3. Delete row i and column j.

Step 4. Set k¼ kþ 1.

Step 5. If k> n, stop, otherwise go to Step 1.

The idea of the above procedure is to restart the algorithm not only in a different

region of the solution space, but also in a region that hopefully contains goodsolutions, thus combining intensification and diversification — which can potentially

guide the TS to a better solution region.

3.7. Tabu tenure sizeTabu tenure is a basis for preventing the search from repeating move performed

in the recent past, potentially reversing the effects of previous moves by interchanges

that might return to previous positions. For example, if we classify as tabu for thethree most recent moves, the tabu tenure size is 3. Tabu tenure size is an important

parameter for TS. Yang et al. (1999) proposed a variable tenure size to be an integer

between n/3 and 3n/2, where n is the problem size. For the proposed FSMP schedul-

ing problem, n is the total number of production batches. Four alternative tenuresizes were set for this research: 0.6n, 0.9n, 1.2n and 1.5n. The second scenario from

table 1 is used to test the performance of the four tenure sizes. The results are shown

in figures 4 and 5 for TS1 and TS2, respectively. The empirical results suggest the

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1 51 101 151

Number of iterations

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0.6n 0.9n 1.2n 1.5n

Figure 4. TS1 Performance versus four Tabu tenure sizes.

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tabu tenure-sizes for TS1 and TS2 are 1.2n and 0.9n, respectively. This set-up is thenused hereafter.

3.8. Stopping criteriaThe stopping criterion is the maximum number of search iterations. For a given

sequence, the total number of possible swaps is equal to n(n� 1)/2. The presentauthors feel that the number of iterations should be a multiple of the totalnumber of possible swaps. As a result of a consideration of figures 4 and 5, thestopping criteria for TS1 and TS2 were both set as n(n� 1). Given the followingnotation, the overall TS procedure is illustrated in figure 6:

T tabu tenure size,Mcount stopping criterion (the maximal number of search iterations),count index of search iteration,Z* current best solution,X* best solution,Xnow current solution,Xnext neighbourhood solution.

In figure 6, the ‘restart’ module uses the EDD rule and intensification–diversification long-term memory to determine the scheduling sequence for itsnext iteration for TS1 and TS2, respectively. In other words, TS1 uses the samerestart sequence (as determined by EDD), whereas TS2 has a changing one (asdetermined by the long-term memory). Due to the inherent random moves ofTS, TS1 can still have improved solutions given the same restart sequence as thesearch progresses. This restart strategy may impose a constraint to the searchalgorithm as contrast to TS2. In general, the long-term memory strategy ismore robust to obtain a quality solution. It may have a trade-off betweenthe increased memory size and improved solution quality depending on itsapplication areas.

4. Empirical illustrations

The five production scenarios in table 1 are used for the empirical illustrations.A steepest descent pairwise interchange (SDPI) heuristic is used to solve the

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Number of iterations

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0.6n 0.9n 1.2n 1.5n

Figure 5. TS2 Performance versus four Tabu tenure sizes.

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problem for benchmarking the quality of the solution against the proposed meth-odologies. For a detailed discussion of the SDPI procedure, see Tompkins et al.(1996). Since the different due-date settings will influence the resulting performancemeasure, there is a need to collect more testing samples to illustrate the effective-ness and efficiency of the proposed methodologies. We decided to collect fivesamples through some preliminary studies. Thus, for each production scenario,

Define Tabu tenure,Mcount, Count=0

Select an initialsolution X now

Count=Count+1 Restart

Let X *=X now, Z*=Z(X now)

Randomly make a non-tabu move then

get X next

Xnow=Xnext,Count=Count+1Select the best X next

in Tabu list

Update tabuattributes

Stop

No

Yes

No

No

No

No

Yes

Yes Yes

Yes

Search all moves inTabu list

All moves not in tabu list have beensearched?

Aspiration criterion satisfied?

Count>Mcount?

Z(X next) better thanZ(X now)?

X now betterthan X *?

Figure 6. The proposed TS procedure.

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five different due-date settings are generated to illustrate the robustness ofthe proposed solution methods. The distributions in the last column oftable 1 are used for the due-date generation. The solution by the simple EDDrule is also collated. Tables 2 and 3 show the performance of the proposedmethodologies.

Table 2 shows the resulting performances from different sequencing methods.Table 3 shows the performance improvement between different methods. Bothtables 2 and 3 show that the TS algorithms consistently outperform the other twoalgorithms, EDD and SDPI. There is not an obvious difference in performancebetween TS1 and TS2. The solution quality by TS is significantly better than thatby SDPI. The results also show that the simple EDD rule cannot provide a qualitysolution, despite its ease of implementation.

Table 4 and figure 7 show the computational efficiency of the proposed TSmethodologies that require less computing time than the time for SDPI for mostinstances. The longest TS computing time is 1549 s (i.e. 25.82min) for the largest

Scenarionumber

SDPI versusEDD

TS1 versusSDPI

TS2 versusSDPI

TS2 versusTS1

1 22.36 39.77 36.87 �4.822 14.52 79.88 68.15 �58.263 19.11 53.63 65.81 26.274 9.33 50.16 53.61 6.935 30.05 43.86 37.69 �10.99

Data are percentages.

Table 3. Summary of tardiness improvement.

Average tardiness (min)

Scenario number EDD SDPI TS1 TS2

1 3744.93 2907.70 1751.44 1835.772 2929.88 2504.44 504.02 797.673 2642.37 2137.33 991.07 730.704 491.10 445.30 221.96 206.575 1294.12 905.29 508.25 564.12

Table 2. Summary of tardiness performance.

Average computational time (s) Ratio

Scenario number (1) SDPI (2) TS1 (3) TS2 (2)/(1) (3)/(1)

1 107 109 109 1.02 1.012 335 254 266 0.76 0.793 883 632 718 0.72 0.814 1106 1031 974 0.93 0.885 2051 1549 1335 0.76 0.65

Table 4. Summary of computational time for the tardiness measure.

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problem size (production scenario 5). In general, the proposed TS methodologiesare quite efficient in solving a practical size problem. However, their efficiency canbe expected to deteriorate as problem size increases.

To gain further insights into the proposed case study, we used flow time (ormakespan) as another performance measure to test the proposed methodologies.This experiment kept the same tabu parameters set-up and the same productionscenarios used in the earlier experiments. The summary of flow time performancemeasure is shown as tables 5 and 6. Table 7 and figure 8 show the summary ofcomputational time.

The experiment for the flow time measure also illustrated the effectivenessand efficiency of the proposed methodologies. Accordingly, it shows promise

0

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1500

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2500

1 2 3 4 5

Scenario

Tim

e(se

c) SDPI

TS1

TS2

Figure 7. Trend of computational time for tardiness measure.

Flow time (min)

Scenario number EDD SDPI TS1 TS2

1 22 404.75 21 798.56 19 362.52 19 126.912 31 690.67 30 301.57 26 168.12 26 465.113 35 842.92 34 271.44 30 898.63 30 063.394 44 105.79 42 392.48 36 341.82 36 470.895 51 219.19 48 566.98 42 569.32 42 841.55

Table 5. Summary of flow time performance.

Scenarionumber

SDPI versusEDD

TS1 versusSDPI

TS2 versusSDPI

TS2 versusTS1

1 2.71 11.18 12.26 1.222 4.38 13.64 12.26 �1.133 4.38 9.84 12.28 2.704 3.88 14.27 13.97 �0.365 5.18 12.35 11.79 �0.64

Data are percentages.

Table 6. Summary of flow time improvement.

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for the proposed methodologies in solving a complex FSMP sequencingproblem.

5. Conclusions and future research opportunities

The present paper has presented a TS simulation optimization approach tosolving an FSMP scheduling sequence problem. A case study from an MLCCmanufacturing system was used to illustrate the proposed methodologies.Empirical results showed promise for the capacity of the proposed methodologiesto solve a practical application. Accordingly, it would seem that the proposed meth-odologies could potentially solve many other planning and control problems inFSMP-type manufacturing systems. Although the scheduling issue was solved inthis research, the dispatching decision was not addressed here, and this might repre-sent an opportunity to improve the quality of the solution. A future research oppor-tunity would be to investigate the impact on the FSMP problem from thedispatching decision onwards. An integrated scheduling and dispatching decisionis another issue that is worthy of investigation. Finally, the link between a shopfloor control system and the proposed decision module would be an important steptowards the implementation of a real-time planning and control system. This repre-sents another future research direction.

Appendices

The appendices contain the data for the case study. Appendix A1 collects infor-mation on the machine inventory, set-up time and downtime. Appendix A2shows lot-size information. Appendices A3–5 collect the detailed processing-timeinformation.

0

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2000

1 2 3 4 5Scenario

Tim

e(se

c) SDPI

TS1

TS2

Figure 8. Trend of computational time for flow time measure.

Average computational time (s) Ratio

Scenario number (1) SDPI (2) TS1 (3) TS2 (2)/(1) (3)/(1)

1 121 74 116 0.61 1.962 303 258 256 0.85 0.843 696 609 614 0.88 0.884 1021 995 997 0.97 0.985 1828 1636 1554 0.89 0.85

Table 7. Summary of computational time for flow time measure.

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A1. Basic data

A2. Lot-size data

The notation for product size and material types is a convention of the case study

company. To protect proprietary information, the company’s definitions are not

provided.

A3. Material-type-dependent process-time data

The process time for the following stages is dependent on material type, regard-

less of product size. The unit of measure is minute-per-lot.

Stage name

Material types

K30 K60 K90 K3000 K3400 K12000

Stage 1: stacking 393.00 555.85 694.80 296.73 446.58 823.60Stage 2: pressing 39.03 44.05 44.05 39.05 39.05 22.05Stage 3: cutting 132.00 176.16 220.20 147.00 147.00 344.88Stage 4: binder burn out 2714.17 2114.17 2534.17 3434.17 2714.17 3434.17

No. Stage nameNumber ofmachines Set-up time (min) MTBF1 (min) MTTR2 (min)

1 Stacking 17 uniform (27, 33)* Expo (30) Expo (5)2 Pressing 1 0 0 03 Cutting 4 uniform (25, 33)y Expo (2) Expo (0.17)4 Binder burn out 16 0 0 05 Sintering 2 constant (1440)* 0 06 Tumbling 8 0 Expo (54) Expo (15)7a Dipping 4 uniform (29, 35)y Expo (25) Expo (3)7b Curing 4 constant (1440)* Expo (35) Expo (2)8 Plating 1 uniform (1, 2)*y Expo (72) Expo (7)9 Testing 12 uniform (20,29)y Expo (2) Expo (0.17)

*Set-up needed for the material change.ySet-up needed for the size change.1Mean time between failures.2Mean time to repair.

Lot size (� 104)

Material types

K12000 K3400 K3000 K90 K60 K30

Product sizes0402 149.5 89.7 77.5 114.4 131.3 95.20603 79.3 41.6 39.6 67.6 53.3 45.30805 31.0 16.0 13.0 26.0 21.0 18.11005 21.7 9.6 7.4 21.9 15.1 12.0

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A4. Product-type-dependent process-time data

The process time for the following stages is dependent on product type. The unit

of measure is minute-per-lot.

A5. Single process-time dataThe process time for the following stages is independent of product-type. The

unit of measure is minute-per-lot.

Acknowledgements

The authors thank the anonymous company for providing the case study.Work was supported in part by the National Science Council of Taiwan, Republicof China, under Grant NSC91-2622-E006-035-CC3.

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Stage 6: tumbling0402 399.17 361.90 374.67 555.83 525.00 690.000603 489.17 459.17 450.00 689.17 689.17 749.170805 541.82 543.67 450.00 753.83 772.50 772.501005 579.17 603.67 483.67 818.83 872.50 832.50

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